Exercise
How many automorphisms does Q(
2) have?
1
2
3
more than 3
No. How about the map a + b2 ->
a - b2?
2 ->
a - b2?
Yes, as follows from the example, every automorphism
f of Q(2) fixes all elements
of Q. The element x = 2 satisfies the
equation x2 = 2.
So f(x) satisfies f(x)2 =
f(x2) = 2. This implies that f(x)
equals 2 or -2.
Thus there are exactly two choices for f.
2) fixes all elements
of Q. The element x = 2 satisfies the
equation x2 = 2.
So f(x) satisfies f(x)2 =
f(x2) = 2. This implies that f(x)
equals 2 or -2.
Thus there are exactly two choices for f.
No way.